In this paper we considered the theory of a symmetric tensor field \(A_{\mu \nu }(x)\)  in 4D, characterized by a particular infinitesimal diffeomorphism transformation, namely that whose vectorial parameter is a 4-divergence of a scalar. This weaker transformation identifies as the most general invariant action the local functional \(S_{inv}\)  (REF ), which depends on a dimensionless parameter, which we called “\(a\) ”, and which cannot be reabsorbed by a redefinition of \(A_{\mu \nu }\) . This is unusual for a quadratic theory and, to our knowledge, unique in 4D, the most known example of a quadratic theory depending on a true parameter being Maxwell Chern Simons theory in 3D, where the parameter plays the role of topological mass for the gauge vector field. This opens the question of the interpretation of the \(a\) -parameter, besides the obvious fact of tuning the invariant action which coincides with LG when \(a=0\)  to suppress gravitons. The fact of dealing with a scalar gauge parameter suggests a gauge fixing governed by a Nakanishi-Lautrup scalar multiplier implementing a gauge condition which, as customary for symmetric rank-2 tensor fields [1]}, [2]}, depends on two gauge parameters. We show how the Landau gauge choice is the natural one and that the value \(a=0\) , which would correspond to pure LG, is forbidden for two reasons. A technical one (the propagator diverge at \(a\rightarrow 0\) ) and a physical one (wrong counting of the degrees of freedom, which turn out to be nine instead of six, as expected in LG). We interpreted this mismatch with the fact that the symmetry of pure LG is governed by a vector gauge parameter (REF ) rather than a scalar one as in (REF ). As a consequence, the gauge fixing needs a vector Nakanishi-Lautrup multiplier as done in [3]}, [1]}, [2]}, [6]}, [7]} and not a scalar one as in (REF ). A final, important comment concerns the relation between the theory studied in this paper and the fracton excitations. The first link is of course given by the symmetry (REF ), which almost defines the fracton theory [8]}, but does not uniquely fixes the fractons as the only quasiparticles of the theory, but rather a mixed phase of fractons and gravitons. What is really peculiar of fractons is the absence of mobility [9]}, [10]}, [11]}. Now, if fractons are localized, we may propose a different (and complementary) interpretation of the \(a\) -parameter. In Section (2.3) we noticed that the case \(a=1\)  we can write the invariant action in terms of the vector field (REF )
\({\cal A}_\rho (x)\equiv \partial _\rho A(x)- \partial ^\nu A_{\nu \rho }(x)\) 
